How Do I Find The Perfect Square Trinomial Whose First Terms Are Q^2+3q?

(q+a)² = q² + 2aq + a²

You have 3q and you need to find a value for a so that
3q = 2aq
3 = 2a
3/2 = a

So,
(q + 3/2)² = q² + 3q + 9/4
_____
This is called "completing the square". It is the basis of a recognized method of solving a quadratic equation, and is the basis of the quadratic formula.